1 | include "basics/jmeq.ma". |
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2 | include "basics/types.ma". |
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3 | include "basics/list.ma". |
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4 | |
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5 | notation > "hvbox(a break ≃ b)" |
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6 | non associative with precedence 45 |
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7 | for @{ 'jmeq ? $a ? $b }. |
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8 | |
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9 | notation < "hvbox(term 46 a break maction (≃) (≃\sub(t,u)) term 46 b)" |
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10 | non associative with precedence 45 |
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11 | for @{ 'jmeq $t $a $u $b }. |
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12 | |
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13 | interpretation "john major's equality" 'jmeq t x u y = (jmeq t x u y). |
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14 | |
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15 | lemma eq_to_jmeq: |
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16 | ∀A: Type[0]. |
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17 | ∀x, y: A. |
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18 | x = y → x ≃ y. |
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19 | // |
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20 | qed. |
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21 | |
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22 | definition inject : ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ λA,P,a,p. dp … a p. |
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23 | definition eject : ∀A.∀P: A → Prop.(Σx:A.P x) → A ≝ λA,P,c.match c with [ dp w p ⇒ w]. |
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24 | |
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25 | coercion inject nocomposites: ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ inject on a:? to Σx:?.?. |
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26 | coercion eject nocomposites: ∀A.∀P:A → Prop.∀c:Σx:A.P x.A ≝ eject on _c:Σx:?.? to ?. |
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27 | |
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28 | (*axiom VOID: Type[0]. |
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29 | axiom assert_false: VOID. |
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30 | definition bigbang: ∀A:Type[0].False → VOID → A. |
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31 | #A #abs cases abs |
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32 | qed. |
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33 | |
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34 | coercion bigbang nocomposites: ∀A:Type[0].False → ∀v:VOID.A ≝ bigbang on _v:VOID to ?.*) |
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35 | |
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36 | lemma sig2: ∀A.∀P:A → Prop. ∀p:Σx:A.P x. P (eject … p). |
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37 | #A #P #p cases p #w #q @q |
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38 | qed. |
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39 | |
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40 | lemma jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. x≃y → x=y. |
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41 | #A #x #y #JMEQ @(jmeq_elim ? x … JMEQ) % |
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42 | qed. |
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43 | |
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44 | coercion jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. ∀p:x≃y.x=y ≝ jmeq_to_eq on _p:?≃? to ?=?. |
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45 | |
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46 | (* END RUSSELL **) |
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