If customer does not get a discount on a product in Cart and product seems to have normal price and there’s no notice under the product that it already has a special price, then find this product in Magento Admin, switch to proper store view (for example TT11 France/France/Francais) and see if this product has Special Price and maybe also special price validity dates that are in the past already. If it does, remove Special Price, Special Price From Date, Special Price To Date and Promotion Type (select empty value).
Do not forget to save product!
Also, make sure, that changes are made in Default view for main site. Set From and To dates for special price to make it visible.